How much more are we talking here? 5 grams? 10 grams? Half the bullet weight in filler?
Unless you do something really, really tricky with shell design and fabrication, like the German Mine shell, high velocity HE shells carried about 10% explosives to weight. Give or take a few percent.
Depends on the velocity range of the projectile and strength/quality of the steel. Lower velocity shells can use thinner walls with more volume for payload. Higher quality steel can allow for thinner walls.
However you can't quite scale a 20mm design down to 13mm exactly. The wall thickness has to stand up to the firing stresses or the shell may buckle while firing. There is also the problem of keeping the shell together in flight due to centrifugal force. A gun with a velocity of 800m/s and a 1 in 400mm twist is going to spin the bullet at 2000rps. Not 2000rpm but 2000rps which is
120,000rpm. You want the shells to reach the target and not disintegrate after leaving the muzzle. The wall thickness in greater in proportion than the 20mm or larger shell. Note that an 700m/s gun has less of a problem? There is also a problem with the fuses. The 13mm fuse can be a little smaller but not scaled exactly in proportion.
The US M23 incendiary bullet (no fuse) .50 cal bullet that was used for troop trials at the end of the war contained 5.8 grams of incendiary in a 32.4 gram bullet instead of the 43-46 gram projectiles. HE and Incendiary material are a lot less dense than steel or copper and light weight bullets are going to have rather different flight characteristics than the other projectiles (different aiming point). Basically you are going to max out at around 3 grams of filler for 13mm projectile.
A five gun setup like that isn't entirely out of the picture, again look at some Italian designs. That's also the layout the SAAB 21 used as well, so it's at the very least worth considering.
I would leave the SAAB 21 out of this. 239sq ft wing, empty weight 3250kg and loaded weight of 4150kg (clean?) and using a DB 605 engine that weighed 725kg (?).
Rather larger and heavier than the French fighters?
And look at the Italian fighters a little more closely. The 12.7mm guns are heavy but the ammo is about 70% as heavy as the French 13.2 mm per round.
The German cannon is lighter than it appears in most lists. It is ready to go at 42 kg, a 42kg Hispano V still needs a 6-8kg belt feeder or a heavy drum if you don't use the belt feed.
Italians were still doing pretty good.
The example I used of the 405 is more about the potential to lighten the basic Hispano instead of the 405 itself. Although I do think the 405 could be a reasonable offensive gun, a lighter 404 or some derivative is more probable and would likely show up earlier than the British Mk V.
Pointing out that while a lighter 404 was certainly possible, it could not be done without sacrificing something. Cutting 40cm off the barrel may not have affected velocity by much and may have helped speed up the rate of fire. Making some of the other moving parts lighter also speeded up the rate of fire. But it also made them less durable/reliable and it might have been a very good trade-off.